ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Negative stationary Killing density in the Kerr ergoregion

Physical scope

A local sign calculation for the canonical scalar stress tensor and stationary Kerr multiplier; it makes no stability or observational claim.

Example

At an exterior point $p$ where the stationary Killing field $T = \partial_{t}$ is spacelike, let $U$ be an oriented coordinate neighborhood of $p$ , restrict the real scalar-field model to $U$ , set its potential to $V = 0$ , fix a coupling $K > 0$ , and let $n$ be any observer at $p$ . There is a smooth scalar field $φ : U \to R$ for which the associated Killing-multiplier density $ρ_{T} (p; n)$ is negative. Thus the density $ρ_{T} (p; n)$ is not nonnegative on all scalar first-derivative data at $p$ , so this multiplier cannot provide a pointwise nonnegative density for all such data there. If a scalar field on $U$ satisfies its scalar equation, its current is conserved. This example does not assert that an on-shell field exists with the prescribed derivative.

Facts & Assumptions

Given: An exterior Kerr point $p$ in the ergoregion, an oriented coordinate neighborhood $U$ of $p$ , an observer $n$ at $p$ , and a smooth real scalar field $φ : U \to R$ in the scalar model with zero potential and positive constant coupling.

[F1]

Kerr horizon angular velocity and the ergoregion: $T = \partial_{t}$ is spacelike when $r_{+} < r < M + \sqrt{M^{2} - a^{2} \cos^{2} θ}$ .

[F2]

Observers, rest spaces and the spatial projection at an event and The observer rest-space decomposition and the spatial projection: $g (n, n) = - 1$ , $n$ is future-directed, and $n^{⊥}$ is positive definite.

[F3]

The scalar-field matter model of the classical gravitational framework: for $V = 0$ the candidate stress tensor is $Q_{a b} = K (\partial_{a} φ \partial_{b} φ - \frac{1}{2} g_{a b} g^{c d} \partial_{c} φ \partial_{d} φ)$ , with $K > 0$ .

[F4]

Killing-multiplier current and its observer density: $j_{T}^{a} = - Q^{a b} T_{b}$ and $ρ_{T} (p; n) = - g (n, j_{T}) = Q_{a b} n^{a} T^{b}$ .

[F5]

First variation, Hilbert tensor and conservation of the scalar-field model: for $V = 0$ , $\nabla^{a} Q_{a b} = K □_{g} φ \partial_{b} φ$ , so $□_{g} φ = 0$ implies $\nabla^{a} Q_{a b} = 0$ .

[F6]

Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor: a symmetric divergence-free tensor contracted with a Killing field gives a divergence-free current.

[A1]

The smooth Lorentzian, observer, and tangent-bundle conventions use only countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

1.1givenA1F1F2algebra

Because $T$ is spacelike by [F1], decompose it relative to $n$ as $T = α n + w$ , where $α : = - g (T, n)$ and $w : = T + g (T, n) n \in n^{⊥}$ . By [F2], if $β : = \sqrt{g (w, w)}$ then $β > ∣ α ∣$ ; set $e : = w / β$ , so $g (e, e) = 1$ , $g (n, e) = 0$ , and $T = α n + β e$ .

2.1F2step 1.1construct

Set $λ : = - (n^{♭} + e^{♭})$ at $p$ , where $n^{♭} = g (n, \cdot)$ and $e^{♭} = g (e, \cdot)$ . Then $λ (n) = 1$ , $λ (e) = - 1$ , and $λ$ vanishes on the orthogonal complement of $span {n, e}$ . In the chosen chart $x : U \to x (U)$ , define $φ : U \to R$ by $φ (q) : = \sum_{a = 0}^{3} λ_{a} (x^{a} (q) - x^{a} (p))$ . This smooth local scalar field has $d φ_{p} = λ$ . Thus $g^{- 1} (λ, λ) = - 1 + 1 = 0$ and $λ (T) = α - β$ .

3.1F3F4step 1.1step 2.1algebra

By [F3], at $p$ the scalar stress tensor is $Q_{a b} = K (λ_{a} λ_{b} - \frac{1}{2} g_{a b} g^{- 1} (λ, λ))$ . Using [F4] and $g^{- 1} (λ, λ) = 0$ , $ρ_{T} (p; n) = Q_{a b} n^{a} T^{b} = K λ (n) λ (T) = K (α - β) < 0$ . Hence the stationary Killing-multiplier density is not nonnegative on all smooth scalar first-derivative data at $p$ .

4.1F5F6step 3.1∎

For any field that additionally satisfies $□_{g} φ = 0$ , [F5] makes $Q$ divergence-free and [F6] makes the associated Killing current divergence-free. This is conditional; the example does not assert existence of an on-shell solution with the selected first derivative. Step 3.1 proves that this multiplier density is not nonnegative on all smooth scalar first-derivative data at $p$ .

Remarks

The source literature also exhibits negative stationary Killing energy for particles in the ergoregion. The calculation above is for the declared massless scalar stress model and does not imply a superradiant instability or any black-hole stability conclusion.

Negative stationary Killing density in the Kerr ergoregion

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources